# Notation for adding and taking away subsets in a family of subsets

Let $$S:=\{A_1,A_2,\ldots,A_k\}$$ for $$A_i\subset\{1,\ldots,n\}$$. I'm trying to figure out the correct notation to refer to $$S_1:=\{A_1,A_2,\ldots,A_k,A_{k+1}\}$$ and $$S_2:=\{A_1,A_2,\ldots,A_{k-1}\}$$. Is $$S_1=S\cup{A_{k+1}}$$ and $$S_2=S\setminus{A_k}$$ or rather $$S_1=S\cup\{A_{k+1}\}$$ and $$S_2=S\setminus\{A_k\}$$? I'm just unsure because the $$A_i$$ are subsets and not elements of $$\{1,\ldots,n\}$$. Also, if I wanted to denote the family $$\{\{1,2\},\{1\}, \{2\},\ldots,\{n\}\}$$ would it be $$\{1,2\}\cup\left(\bigcup_{1\leq{i}\leq{n}}\{i\}\right)$$ or $$\{\{1,2\}\}\cup\left(\bigcup_{1\leq{i}\leq{n}}\{\{i\}\}\right)$$ or something else? Thanks.

Your second assumption where we had $$S_1 = S \cup \{A_{k+1}\}$$ and $$S_2 = S \setminus \{A_k\}$$ is correct. Sure, the $$A_i$$ are sets themselves, but $$S_1$$ and $$S_2$$ are sets of sets. If we want to add or remove elements to (or from) the $$S_i$$, we also need to add or remove a set that contains sets (the $$A_i$$) as its elements. A similar logic would be true for your second question. You would want $$\{\{1,2\}\} \cup \left(\bigcup_{i=1}^n \{\{i\}\} \right)$$

Your second choices are correct in all three cases: $$S_1 = S \cup \{A_{k+1}\} \\ S_2 = S \setminus \{A_k\} \\ \{\{1,2\},\{1\}, \{2\},\dots,\{n\}\} = \{\{1,2\}\}\cup\left(\bigcup_{i=1}^n\{\{i\}\}\right)$$ But I think the versions with $$\dots$$ are the clearest. Forget about $$S,S_1,S_2$$, and instead define $$S_k := \{A_1,A_2,\dots,A_k\}$$

• Thanks. It's just that in my paper, $S$ is already defined and my proof requires a modification to one of the $A_i$ where $i$ may vary and is not necessarily $k$ in all cases.
– Ari
Feb 4, 2021 at 15:21